Example 3.3 Consider configurations of ordered tuples of \(n\ge 4\)  points on \(\mathbb {P}^1\)  up to \(\mathrm {SL}_2\) -action, with the natural linearisation and the Killing form on \(\mathrm {SL}_2\) . Such a configuration is semistable as long as no more than \(n/2\)  of the points coincide anywhere, and [1]} shows that the HKKN stratification measures how many points do coincide. In this case the above categorical quotient given above is most unsatisfactory: for any \(\beta \)  it is simply a point; see [2]} for more details.

